Question

The minimum value of the polynomial x (x + 1)(x + 2)(x + 3) IS
(a) 0
(c)-1​

Answer 1

Given : polynomial x (x + 1)(x + 2)(x + 3)

To find :  minimum value of the polynomial

Solution:

P(x) = x (x + 1)(x + 2)(x + 3)

=> P(x) = x(x + 3)(x + 1)(x + 2)

=> P(x) = (x² + 3x )(x² + 3x + 2)

=> P(x) =  (x² + 3x )² + 2(x² + 3x)

Let say y = x² + 3x

p(y) =   y²  + 2y

p'(y) = 2y  +  2

=> y = - 1

p''(y) = 2  > 0

=> y = -1  will give minimum value

p(y) =   y²  + 2y

=> P(-1) = (-1)² + 2(-1)

=> P(-1) = 1  - 2

=> P(-1) = - 1

Hence minimum value of the polynomial x (x + 1)(x + 2)(x + 3) is  - 1

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